I have an equation for determining the acceleration of an object being propelled by a constant power source, that is affected by air resistance:(adsbygoogle = window.adsbygoogle || []).push({});

a = [itex]\frac{P}{mv}[/itex]-[itex]\frac{C_{D}pAv^{2}}{2m}[/itex]

Since F = [itex]\frac{P}{v}[/itex]

I am trying to graph this as a velocity-time graph, however, I don't know how to do it. There is no time variable that I can replace with x, and the y-value (velocity) is mixed into the equation already.

I remember the equation given to me for a similar sort of thing, without air resitance, but instead just a constant friction force, that was something like this:

y = k(1-e^{-ax})

Where k is a constant, which is the maximum speed, and a is another constant which represents the force of air resistance.

The maximum speed in this case is [itex]\sqrt[3]{\frac{2P}{C_{D}pA}}[/itex], so the equation would be something like:

y = [itex]\sqrt[3]{\frac{2P}{C_{D}pA}}[/itex](1-e^{-ax})

But that's as far as I've gotten. By the use of iteration, I have determined the velocity at several different times, here's a few, just in case it helps:

(0,0) (25,83.4762) (50,118.1195) (75,126.1601) (100,127.7624) (125,128.0709)

This is for variables of values: p = 0.001, A = 1900.933, m = 1900.933, P = 1*10^{6}

and C_{D}= 0.5

Any help would be greatly appreciated, and if you need any more information, just let me know.

EDIT: the equations don't seem to be formatting correctly, so I'll redo them down here:

a = P/v - (Cd*p*A*v^2) /2

F = P/v

max speed = ( (2*P) / (Cd*p*A) )^1/3

y = ( (2*P) / (Cd*p*A) )^1/3 * (1 - e^-ax)

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# Finding the graph formula with know points and other equation.

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